Determine how many solutions exist for the system of equations. ${-6x+y = 7}$ ${2x+2y = -18}$
Solution: Convert both equations to slope-intercept form: ${-6x+y = 7}$ $-6x{+6x} + y = 7{+6x}$ $y = 7+6x$ ${y = 6x+7}$ ${2x+2y = -18}$ $2x{-2x} + 2y = -18{-2x}$ $2y = -18-2x$ $y = -9-x$ ${y = -x-9}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 6x+7}$ ${y = -x-9}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.